How are great circle tracks and rhumb lines drawn when not on a chart?

[Red123]

When answering questions I have noticed that they all take the rhumb line track as being straight and the great circle track as being curved towards the equator. Why is this the case?

In my G-nav atpl book (Padpilot), and online, I have found sources which state that the rhumb line is curved and the great circle straight. So why in the questions do they take them as being the other way round when they aren't on a chart?


For example, 'what is the initial great circle direction from 45*N14*12'W to 45*N12*48'E?' takes the rhumb line as straight and the great circle as a curve. Whereas, 'an aircraft flies from point A at 44*54.0'N33*W to point B at 44*54.0'N052*E by following a great circle route. The initial direction at point A is 060*. What is the approximate direction of the route at point B' draws the great circle as a straight line and the rhumb line as a curve?


Please can someone explain why the questions do this? I've wasted a long time trying to figure it out and guess I must be missing something obvious!

[ersa]

I have just started reading GNAV ATPL and it makes no sense either....

From my notes great circles are generally straight lines the shortest distance between 2 points...a rhumb line is a line of constant direction, the longest between 2 points .

This will change depending on the conformity of the chart..
conic projection, mercators projection, etc

Thats as far as I have read

[Alex Whittingham]

A great circle track is the one you would get if you stretch a rubber band across the globe from point to point. A rhumb line would lie on the equatorial side of that and be a longer (constant direction) track.

These lines exist on a spherical surface. You can't represent them totally accurately on a piece of paper, only approximate to them. The purpose of the sketch diagrams we draw is only to get the sense of the question right, and to determine whether perhaps convergency is added or subtracted, or conversion angle.

For this reason I would suggest you lay out your diagrams in a standard style, and make no attempt to get them looking exactly like the lines would look on the earth's surface, because if you do you are going to fail anyway. The style I use follows these rules:

1. Determine which hemisphere you are in, draw in the meridians as straight lines with an easily discernible slope that reflects the hemisphere.
2. Draw in a representation of the great circle track as a straight horizontal line between the two meridians. Make sure it extends beyond the meridians.
3. Look at the question. Decide whether the track is generally Easterly or generally Westerly. If Easterly your start point is on the left and your end point on the right, vice versa if Westerly. Label points 'A' and 'B' or whatever.
4. Look at the question. If there is a rhumb line involved draw it in as a curved track on the equatorial side of the straight line great circle track between points A and B. Make sure it extends out beyond points A and B.

There are two possible reasons why someone would draw the rhumb line as a straight line and the great circle as a curve. The first and most common one is that they are confused and under confident about exactly what great circles and rhumb lines are (and who isn't when they first meet them??) and the second is when someone is trying to represent how the lines appear on a Mercator chart, which is a very particular archaic chart projection which does actually show rhumb lines as straight lines and great circles as curves. If you are asked questions on a Mercator you can draw the diagram out as it would be on the chart but the question will work just as well if you stick to a 'standard diagram' format as above.

I will try to add an example question with a diagram to this thread later today. Red, if you could provide the full text of your first example I will do that one for you.

[RichardH]

Alex is correct, pictures are essential to get this correct, I just wish PPrune made this easier!

Over the Earth GCs are shortest routes with changing direction and RLs are of constant direction. GCs ALWAYS lie polar side of RLs. However it is easier to draw them as Alex described.

The other thing to spot in the first example N45 W014 12 to N45 E012 48 is the two latitudes are the same this means that the RL must either be 270 or 090 as in this case as you are going from W to E.

Then you are into convergency = Ch Long * Sine Mean Lat. Two different hemispheres means add them to get 37 degrees Ch Long * Sine 45 = 26 degrees. The angle between your RL of 090 and the GC is HALF convergency = 13 (conversion angle). RL of 090 - 13 gives you INITIAL GC track (at W014 12) of 077 and FINAL GC track (at E012 48) of 103.

As said much easier with diagram.

[Alex Whittingham]

With due respect to your experience, Richard, I'm going to disagree! First of all, though, I would like to define what I mean by straight, and this would be 'direct, non-deviating, not bent'. By this definition a great circle is a straight line. It also meets the requirement 'a straight line is the shortest distance between two points', albeit on a spherical surface.

Rhumb lines are not straight lines on the spherical surface nor on any chart projection except the Mercator. Was that a typo?

[Alex Whittingham]

this would be my sketch for the above:

https://dl.dropboxusercontent.com/u/26811764/great%20circle.png

[AlphaMale]

Alex's diagram is the easiest way to look at it.

Red123 - I'm wondering if you're getting confused with conic projections?

Rhumb lines are curves concave to the pole. Great circles are taken to be straight but have a small curve that is concave to the parallel of origin.

For the calculation you are doing, this doesn't come into play. Take all great circles as being straight lines.

Good luck with your exams.

[RTN11]

You have to bear in mind that the maps are made using a globe with a light bulb in the middle projecting out, it's how you wrap the paper round the sphere that determines the projection.

You really need to get the very basics of this understood before you can progress on to advanced questions.

Perhaps get a globe and wrap the paper round in the typical conic and Mercator projection styles to see how different a projection they would make.

[keith williams]

The procedure for solving this type of question has been explained by previous posters, so I will confine my post to answering the original question:

How are great circle tracks and rhumb lines drawn when not on a chart?

The two most obvious reasons for making a drawing are:

1. To solve the problem by measuring angles directly from the drawing.

2. To assist in visualize the situation.

Reason 1 applies to a number of exam questions mainly those relate to Polar Stereographic charts.

Reason 2 applies to all other exam questions with the exception of those which refer to a specific type of chart.

The sketch provided by Alex has the following advantages:

1. By showing the meridians as converging lines it makes it obvious which hemisphere (Northern or Southern) is involved.

2. It correctly shows that the Great Circle is a straight line and is shorter than the Rhumb Line.

But it has the disadvantage that it shows parallels of latitude as curved lines, which is contrary to most people’s intuitive vision.

An alternative sketch also shows the meridians as converging lines but shows the Rhumb Lines as straight lines, Great Circles are shown as curved lines. This has the following advantages:


1. By showing the meridians as converging lines it makes it obvious which hemisphere (Northern or Southern) is involved.

2. It shows the parallels of latitude as straight lines, which complies with most people’s intuitive vision.

3. It shows Great Circles as lines which curve up towards the nearest pole and then back towards the Equator. This is how a Great Circle would appear when looking at a side view of a model of the Earth.

Each of the two methods have their advantages, so the choice of which to use is one of personal preference. Both methods will achieve the aim of assisting in visualizing the situation.

The question of whether a Great Circle or a Rhumb Line is a straight line is a curious one. The Equator is a straight line and is a Great Circle because its centre is at the centre of the Earth. But it is also a Rhumb Line because it crosses all meridians at the same angle.

The most commonly used definition of a Rhumb Line is one which crosses all meridians in its path at the same angle. Using this definition, all parallels of latitude and meridians are Rhumb Lines. The meridians are straight-line Rhumb Lines and when coupled with their anti-meridians form great circles. But if we were to draw any Rhumb Line which was not the Equator or a meridian/anti-meridian, it would not produce a circle, but would spiral towards the poles. So those Rhumb Lines which are the Equator or meridians are straight lines, but all other Rhumb lines are curves.

So in attempting to distinguish between Great Circles and Rhumb Lines by testing the straightness, all that can be really said is that all Great Circles are straight lines but some Rhumb Lines are not.

[Alex Whittingham]

Some good points. I do think it is important that the instructors in the industry present as near a unified view as possible on this point because, although 'learned discussion' has its place, it confuses the hell out of the students if they think that even the instructors do not agree whether a great circle is a straight line or not.

Whilst I take your point that some people may have an 'intuitive vision' that rhumb lines are straight lines this does not stand up to inspection and should be countered early on in the training.

Certainly a straight line on a map is not a rhumb line (except on Mercators), it is a very close approximation to a great circle. Sure, we draw a line on a map to fly from A to B but if it has any length at all we can see that track direction at the beginning is not the same as the end. Yes, this might be partly due to changes in variation if you are measuring magnetic tracks but, if you measure true tracks against the meridians you can see the effect of convergency. In light aircraft you might measure the start track as, say, 060, and the end track as 062 and choose to fly 061. In doing this you are actually flying the rhumb line (half the convergency added to the initial GC track) but drawing the great circle on the map. The track displacement is so small you don't notice it in flight, even if you can hold heading to 1° of accuracy.

For me the clearest indication as to which is 'straight' comes when you look at the globe. Yes it is possible to choose a viewing angle where a rhumb line looks straight. Look at this image, at the latitude that passes through southern France.

https://dl.dropboxusercontent.com/u/26811764/side%20view.PNG

The proper way to look at it, though, has to be from directly above, where the rhumb line can be seen to be curved just like it is on all map representations (except Mercators). If you look at great circles from directly above they always look straight just like (their very near approximations) do (within the usable areas) on all charts (except Mercators)

https://dl.dropboxusercontent.com/u/26811764/from%20above.PNG

Consistency alone calls for rhumb lines to be sketched as curves, anything else confuses.

[keith williams]

I do think it is important that the instructors in the industry present as near a unified view as possible on this point because, although 'learned discussion' has its place, it confuses the hell out of the students if they think that even the instructors do not agree whether a great circle is a straight line or not.

We will only achieve that by getting all instructors to say only the things which are true. The statement that “all rhumb lines are curves” is untrue. As explained in my previous post some are curves and some (meridians, and parallels of latitude) are not.

Telling half truths will cause even more confusion. We may for example tell them the following:

1. All parallels of latitude are rhumb lines.
2. All rhumb lines spiral towards the Equator.

Any student with half a brain will then confuse him/her self to hell trying to understand how the parallels of latitude spiral towards the poles.


Whilst I take your point that some people may have an 'intuitive vision' that rhumb lines are straight lines

I did not say that. What I actually said was:

But it has the disadvantage that it shows parallels of latitude as curved lines, which is contrary to most people’s intuitive vision.

And

It shows the parallels of latitude as straight lines, which complies with most people’s intuitive vision.

When people are asked to visualize the globe with the meridians and parallels of latitude, many, probably most, will see the side view with straight horizontal lines representing the parallels of latitude. For these people sketching rhumb lines as straight lines will be easiest. Some people will of course it from a slightly higher angle, with both the meridians and the parallels of latitude represented by curves. Those people may find it easier to sketch rhumb lines as curves. Each student should use the method which works best for him/her.

The proper way to look at it, though, has to be from directly above.

I think that only God (and possibly a very few astronauts now and then) look at the Earth from directly above. There is no universally accepted proper or improper way to view it. The “most proper way to look at it” is whatever way each individual finds to be the intuitive one for them.

[Alex Whittingham]

Ah no, no difficulty, assuming you meant 'the equator and meridians'. Those particular rhumb lines are straight lines because they are great circles. I do disagree with your apparent assertion that parallels of latitude are straight lines and I think that to teach that to students would be (a) wrong and (b) deeply confusing. Another world view, from above.

https://dl.dropboxusercontent.com/u/26811764/above.PNG

Note the parallel of latitude at around 75°N. If you took it to the extreme and looked at the parallel of latitude at 89°59'N you would have a circle with a 1NM radius around the pole. I see you could get down really low and squint at it and suggest that viewed from the side it might be straight but most people would call it a circle.

I appreciate your pedantry, and it is of course important to recognise the exceptions. You know I did not say 'all rhumb lines are curves', I chose my words carefully. The full statement, were I to make it, would be "all rhumb lines are curves on the earth's surface and on map projections (except Mercators) with the exception of rhumb lines which are also great circles in which case they are straight lines on the earth (but on a spherical surface) and nearly straight lines on all map projections used within their usable area (except Mercators)".

This question was not, however, about full definitions, the OP asked how to show them in a sketch, and pedantic bickering does not help, in fact it must put most people off. As I sit at risk of being accused of the same level of pedantry I will rest my argument now.

[keith williams]

Yes Alex you are correct. On reflection I can see that my arguments that parallels of latitude are straight lines were incorrect. I have deleted them to avoid confusing any students reading this thread. To avoid any charges that I have attempted to hide my mistakes I have also stated the reason for the deletions.

All of the other arguments in my posts remain unchanged.


As for your suggestion that I am being pedantic:

1. RichardH stated that Rhumb Lines are straight lines.
2. You stated that they are curves.
3. I stated that some are straight lines and some are curves.

Of the three statements, mine is the most accurate. If that is being pedantic then all I can say is "Guilty as charged your honour."